Differential Equations

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Differential Equations







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Genre: eLearning

Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODEs) deal with functions of one variable, which can often be thought of as time.

Topics include: Solution of first-order ODE's by analytical, graphical and numerical methods; Linear ODE's, especially second order with constant coefficients; Undetermined coefficients and variation of parameters; Sinusoidal and exponential signals: oscillations, damping, resonance; Complex numbers and exponentials; Fourier series, periodic solutions; Delta functions, convolution, and Laplace transform methods; Matrix and first order linear systems: eigenvalues and eigenvectors; and Non-linear autonomous systems: critical point analysis and phase plane diagrams.

LECTURES
Lecture 1 - The Geometrical View of y'=f(x,y): Direction Fields, Integral Curves
Lecture 2 - Euler's Numerical Method for y'=f(x,y) and its Generalizations
Lecture 3 - Solving First-order Linear ODE's; Steady-state and Transient Solutions
Lecture 4 - First-order Substitution Methods: Bernouilli and Homogeneous ODE's
Lecture 5 - First-order Autonomous ODE's: Qualitative Methods, Applications
Lecture 6 - Complex Numbers and Complex Exponentials
Lecture 7 - First-Order Linear with Constant Coefficients
Lecture 8 - Applications to Temperature, Mixing, RC-circuit, Decay, and Growth Models
Lecture 9 - Solving Second-Order Linear ODE's with Constant Coefficients
Lecture 10 - Complex Characteristic Roots; Undamped and Damped Oscillations
Lecture 11 - Second-Order Linear Homogeneous ODE's: Superposition, Uniqueness, Wronskians
Lecture 12 - Inhomogeneous ODE's; Stability Criteria for Constant-Coefficient ODE's
Lecture 13 - Inhomogeneous ODE's: Operator and Solution Formulas Involving Ixponentials
Lecture 14 - Interpretation of the Exceptional Case: Resonance
Lecture 15 - Introduction to Fourier Series; Basic Formulas for Period 2(pi)
Lecture 16 - More General Periods; Even and Odd Functions; Periodic Extension
Lecture 17 - Finding Particular Solutions via Fourier Series; Resonant Terms
Lecture 18 - Derivative Formulas; Using the Laplace Transform to Solve Linear ODE's
Lecture 19 - Convolution Formula: Proof, Connection with Laplace Transform, Application
Lecture 20 - Using Laplace Transform to Solve ODE's with Discontinuous Inputs
Lecture 21 - Impulse Inputs; Dirac Delta Function, Weight and Transfer Functions
Lecture 22 - First-Order Systems of ODE's; Solution by Elimination, Geometric Interpretation
Lecture 23 - Homogeneous Linear Systems with Constant Coefficients: Solution via Matrix Eigenvalues
Lecture 24 - Continuation: Repeated Real Eigenvalues, Complex Eigenvalues
Lecture 25 - Sketching Solutions of 2x2 Homogeneous Linear System with Constant Coefficients
Lecture 26 - Matrix Methods for Inhomogeneous Systems
Lecture 27 - Matrix Exponentials; Application to Solving Systems
Lecture 28 - Decoupling Linear Systems with Constant Coefficients
Lecture 29 - Non-linear Autonomous Systems: Finding the Critical Points and Sketching Trajectories
Lecture 30 - Limit Cycles: Existence and Non-existence Criteria
Lecture 31 - Non-Linear Systems and First-Order ODE's

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pass: sam05_ND

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